Indeterminacy and well-posedness of the non-local theory of Rayleigh waves

The latest literature stance holds that, in a 2D framework, the non-local theory of elasticity, as developed by Eringen, is fundamentally inconsistent because “it does not satisfy the equations of motion for [the] non-local stresses”. In fact, it is believed that the differential form of this theory, that is accessible when the attenuation function is a Green function and that is well-posed, gives different results from the integral formulation. We show that these ideas are ill-conceived, provided that we adopt the kernel modification approach, by which the constitutive boundary conditions (CBCs) embedded in the integral formulation are reconciled with the natural boundary conditions of the problem at hand. Indeed, this kernel modification strategy, which was first introduced by the authors for 1D non-local models, is necessary to avoid that the problem becomes over-constrained through (possibly conflicting) natural and constitutive boundary conditions, and consequently ill-posed. Once the problem is made well-posed, we show that (1) failure to satisfy the equations of motion is not only expected, but it is in fact necessary, (2) for the example case of surface waves propagating in a stress-free half-plane, the integral and the differential formulations coincide, (3) for a force problem, the non-local theory is generally indeterminate because it lacks compatibility: consequently, for a unique solution, an extra boundary condition is needed, and (4) multiple Rayleigh wave branches appear as a consequence of non-locality.